Comment on "On the Uniqueness of Stable Marriage Matchings" [Economic Letters 69(1):1-8, 2000]

We prove with the help of a counterexample that Lemma 6 and Corollary 7 from Eeckhout [1] are incorrect.

Essays on Matching Markets and Their Equilibria

Matching theory and matching markets are a core component of modern economic theory and market design. This dissertation presents three original contributions to this area. The first essay constructs a matching mechanism in an incomplete information matching market in which the positive assortative match is the unique efficient and unique stable match. The mechanism asks each agent in the matching market to reveal her privately known type. Through its novel payment rule, truthful revelation forms an ex post Nash equilibrium in this setting. This mechanism works in one-, two- and many-sided matching markets, thus offering the first mechanism to unify these matching markets under a single mechanism design framework. The second essay confronts a problem of matching in an environment in which no efficient and incentive compatible matching mechanism exists due to matching externalities. I develop a two-stage matching game in which a contracting stage facilitates subsequent conditionally efficient and incentive compatible Vickrey auction stage. Infinite repetition of this two-stage matching game enforces the contract in every period. This mechanism produces inequitably distributed social improvement: parties to the contract receive all of the gains and then some. The final essay demonstrates the existence of prices which stably and efficiently partition a single set of agents into firms and workers, and match those two sets to each other. This pricing system extends Kelso and Crawford's general equilibrium results in a labor market matching model and links one- and two-sided matching markets as well.

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + ε)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.

Matching Markets under (In)complete Information

We are the first to introduce incomplete information to centralized many-to-one matching markets such as those to entry-level labor markets or college admissions. This is important because in real life markets (i) any agent is uncertain about the other agents' true preferences and (ii) most entry-level matching is many-to-one (and not one-to-one). We show that for stable (matching) mechanisms there is a strong and surprising link between Nash equilibria under complete information and Bayesian Nash equilibria under incomplete information. That is,given a common belief, a strategy profile is a Bayesian Nash equilibrium under incomplete information in a stable mechanism if and only if, for any true profile in the support of the common belief, the submitted profile is a Nash equilibrium under complete information at the true profile in the direct preference revelation game induced by the stable mechanism. This result may help to explain the success of stable mechanisms in these markets.

Manipulation via Capacities Revisited

This paper revisits manipulation via capacities in centralized two-sided matching markets. Sönmez (1997) showed that no stable mechanism is nonmanipulable via capacities. We show that non-manipulability via capacities can be equivalently described by two types of non-manipulation via capacities: non-Type-I-manipulability meaning that no college with vacant positions can manipulate by dropping some of its empty positions; and non-Type-II-manipulability meaning that no college with no vacant positions can manipulate by dropping some of its filled positions. Our main result shows that the student-optimal stable mechanism is the unique stable mechanism which is non-Type-I-manipulable via capacities and independent of truncations. Our characterization supports the use of the student-optimal stable mechanism in these matching markets because of its limited manipulability via capacities by colleges.

On the exhaustiveness of truncation and dropping strategies in many-to-many matching markets

We consider two–sided many–to–many matching markets in which each worker may work for multiple firms and each firm may hire multiple workers. We study individual and group manipulations in centralized markets that employ (pairwise) stable mechanisms and that require participants to submit rank order lists of agents on the other side of the market. We are interested in simple preference manipulations that have been reported and studied in empirical and theoretical work: truncation strategies, which are the lists obtained by removing a tail of least preferred partners from a preference list, and the more general dropping strategies, which are the lists obtained by only removing partners from a preference list (i.e., no reshuffling). We study when truncation / dropping strategies are exhaustive for a group of agents on the same side of the market, i.e., when each match resulting from preference manipulations can be replicated or improved upon by some truncation / dropping strategies. We prove that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 1). We show that this result cannot be extended neither to group manipulations (even when all quotas equal 1 – Example 1), nor to individual manipulations when the agent’s quota is larger than 1 (even when all other agents’ quotas equal 1 – Example 2). Finally, we prove that for each stable mechanism, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 2), i.e., independently of the quotas.

Essays on the Axiomatic Theory of Matching

This dissertation mimics the Turkish college admission procedure. It started with the purpose to reduce the inefficiencies in Turkish market. For this purpose, we propose a mechanism under a new market structure; as we prefer to call, semi-centralization. In chapter 1, we give a brief summary of Matching Theory. We present the first examples in Matching history with the most general papers and mechanisms. In chapter 2, we propose our mechanism. In real life application, that is in Turkish university placements, the mechanism reduces the inefficiencies of the current system. The success of the mechanism depends on the preference profile. It is easy to show that under complete information the mechanism implements the full set of stable matchings for a given profile. In chapter 3, we refine our basic mechanism. The modification on the mechanism has a crucial effect on the results. The new mechanism is, as we call, a middle mechanism. In one of the subdomain, this mechanism coincides with the original basic mechanism. But, in the other partition, it gives the same results with Gale and Shapley's algorithm. In chapter 4, we apply our basic mechanism to well known Roommate Problem. Since the roommate problem is in one-sided game patern, firstly we propose an auxiliary function to convert the game semi centralized two-sided game, because our basic mechanism is designed for this framework. We show that this process is succesful in finding a stable matching in the existence of stability. We also show that our mechanism easily and simply tells us if a profile lacks of stability by using purified orderings. Finally, we show a method to find all the stable matching in the existence of multi stability. The method is simply to run the mechanism for all of the top agents in the social preference.

Preference Structure and Random Paths to Stability in Matching Markets

This paper examines how preference correlation and intercorrelation combine to influence the length of a decentralized matching market's path to stability. In simulated experiments, marriage markets with various preference specifications begin at an arbitrary matching of couples and proceed toward stability via the random mechanism proposed by Roth and Vande Vate (1990). The results of these experiments reveal that fundamental preference characteristics are critical in predicting how long the market will take to reach a stable matching. In particular, intercorrelation and correlation are shown to have an exponential impact on the number of blocking pairs that must be randomly satisfied before stability is attained. The magnitude of the impact is dramatically different, however, depending on whether preferences are positively or negatively intercorrelated.

Slow Isotope Turnover Rates and Low Discrimination Values in the American Alligator: Implications for Interpretation of Ectotherm Stable Isotope Data

Stable isotope analysis has become a standard ecological tool for elucidating feeding relationships of organisms and determining food web structure and connectivity. There remain important questions concerning rates at which stable isotope values are incorporated into tissues (turnover rates) and the change in isotope value between a tissue and a food source (discrimination values). These gaps in our understanding necessitate experimental studies to adequately interpret field data. Tissue turnover rates and discrimination values vary among species and have been investigated in a broad array of taxa. However, little attention has been paid to ectothermic top predators in this regard. We quantified the turnover rates and discrimination values for three tissues (scutes, red blood cells, and plasma) in American alligators (Alligator mississippiensis). Plasma turned over faster than scutes or red blood cells, but turnover rates of all three tissues were very slow in comparison to those in endothermic species. Alligator δ15N discrimination values were surprisingly low in comparison to those of other top predators and varied between experimental and control alligators. The variability of δ15N discrimination values highlights the difficulties in using δ15N to assign absolute and possibly even relative trophic levels in field studies. Our results suggest that interpreting stable isotope data based on parameter estimates from other species can be problematic and that large ectothermic tetrapod tissues may be characterized by unique stable isotope dynamics relative to species occupying lower trophic levels and endothermic tetrapods.

QoE-aware power management in vehicle-to-grid networks:a matching-theoretic approach

Frequency, time and places of charging and discharging have critical impact on the Quality of Experience (QoE) of using Electric Vehicles (EVs). EV charging and discharging scheduling schemes should consider both the QoE of using EV and the load capacity of the power grid. In this paper, we design a traveling plan-aware scheduling scheme for EV charging in driving pattern and a cooperative EV charging and discharging scheme in parking pattern to improve the QoE of using EV and enhance the reliability of the power grid. For traveling planaware scheduling, the assignment of EVs to Charging Stations (CSs) is modeled as a many-to-one matching game and the Stable Matching Algorithm (SMA) is proposed. For cooperative EV charging and discharging in parking pattern, the electricity exchange between charging EVs and discharging EVs in the same parking lot is formulated as a many-to-many matching model with ties, and we develop the Pareto Optimal Matching Algorithm (POMA). Simulation results indicates that the SMA can significantly improve the average system utility for EV charging in driving pattern, and the POMA can increase the amount of electricity offloaded from the grid which is helpful to enhance the reliability of the power grid.

Matching theory based travel plan aware charging algorithms in V2G smart grid networks

The frequency, time and places of charging have large impact on the Quality of Experience (QoE) of EV drivers. It is critical to design effective EV charging scheduling system to improve the QoE of EV drivers. In order to improve EV charging QoE and utilization of CSs, we develop an innovative travel plan aware charging scheduling scheme for moving EVs to be charged at Charging Stations (CS). In the design of the proposed charging scheduling scheme for moving EVs, the travel routes of EVs and the utility of CSs are taken into consideration. The assignment of EVs to CSs is modeled as a two-sided many-to-one matching game with the objective of maximizing the system utility which reflects the satisfactory degrees of EVs and the profits of CSs. A Stable Matching Algorithm (SMA) is proposed to seek stable matching between charging EVs and CSs. Furthermore, an improved Learning based On-LiNe scheduling Algorithm (LONA) is proposed to be executed by each CS in a distributed manner. The performance gain of the average system utility by the SMA is up to 38.2% comparing to the Random Charging Scheduling (RCS) algorithm, and 4.67% comparing to Only utility of Electric Vehicle Concerned (OEVC) scheme. The effectiveness of the proposed SMA and LONA is also demonstrated by simulations in terms of the satisfactory ratio of charging EVs and the the convergence speed of iteration.

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in R(k), called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in Rk. We also discuss a simple signaling pathway related to cancer research, called p53 module.

A further note on the college admission game

When a stable matching rule is used for a college admission market, questions on incentives facing agents of both sides of the market naturally emerge. This note states and proves four important results which fill a gap in the theory of incentives for the college admission model. Two of them have never been demonstrated but have been used along the years and are responsible for the success that this theory has had in explaining empirical economic phenomena.

Characterization of an In Vitro Model of Elastic Fiber Assembly

Elastic fibers consist of two morphologically distinct components: elastin and 10-nm fibrillin-containing microfibrils. During development, the microfibrils form bundles that appear to act as a scaffold for the deposition, orientation, and assembly of tropoelastin monomers into an insoluble elastic fiber. Although microfibrils can assemble independent of elastin, tropoelastin monomers do not assemble without the presence of microfibrils. In the present study, immortalized ciliary body pigmented epithelial (PE) cells were investigated for their potential to serve as a cell culture model for elastic fiber assembly. Northern analysis showed that the PE cells express microfibril proteins but do not express tropoelastin. Immunofluorescence staining and electron microscopy confirmed that the microfibril proteins produced by the PE cells assemble into intact microfibrils. When the PE cells were transfected with a mammalian expression vector containing a bovine tropoelastin cDNA, the cells were found to express and secrete tropoelastin. Immunofluorescence and electron microscopic examination of the transfected PE cells showed the presence of elastic fibers in the matrix. Biochemical analysis of this matrix showed the presence of cross-links that are unique to mature insoluble elastin. Together, these results indicate that the PE cells provide a unique, stable in vitro system in which to study elastic fiber assembly.

Necessary conditions for the emergence of homochirality via autocatalytic self-replication

We analyze a recent proposal for spontaneous mirror symmetry breaking based on the coupling of first-order enantioselective autocatalysis and direct production of the enantiomers that invokes a critical role for intrinsic reaction noise. For isolated systems, the racemic state is the unique stable outcome for both stochastic and deterministic dynamics when the system is in compliance with the constraints dictated by the thermodynamics of chemical reaction processes. In open systems, the racemic outcome also results for both stochastic and deterministic dynamics when driving the auto-catalysis unidirectionally by external reagents. Nonracemic states can result in the latter only if the reverse reactions are strictly zero: these are kinetically controlled outcomes for small populations and volumes, and can be simulated by stochastic dynamics. However, the stability of the thermodynamic limit proves that the racemic outcome is the unique stable state for strictly irreversible externally driven autocatalysis. These findings contradict the suggestion that the inhibition requirement of the Frank autocatalytic model for the emergence of homochirality may be relaxed in a noise-induced mechanism.